DLMF:14.13#Ex1 (Q4836)

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DLMF:14.13#Ex1
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    ± 1 2 π i 𝖯 ν μ ( cos θ ) + 𝖰 ν μ ( cos θ ) = π 1 2 Γ ( ν + μ + 1 ) ( 2 sin θ ) μ e ± ( ν + μ + 1 ) i θ 𝐅 ( ν + μ + 1 , μ + 1 2 ; ν + 3 2 ; e ± 2 i θ ) . plus-or-minus 1 2 𝜋 𝑖 Ferrers-Legendre-P-first-kind 𝜇 𝜈 𝜃 Ferrers-Legendre-Q-first-kind 𝜇 𝜈 𝜃 superscript 𝜋 1 2 Euler-Gamma 𝜈 𝜇 1 superscript 2 𝜃 𝜇 superscript 𝑒 plus-or-minus 𝜈 𝜇 1 𝑖 𝜃 scaled-hypergeometric-bold-F 𝜈 𝜇 1 𝜇 1 2 𝜈 3 2 superscript 𝑒 plus-or-minus 2 𝑖 𝜃 {\displaystyle{\displaystyle\pm\frac{1}{2}\pi i\mathsf{P}^{\mu}_{\nu}\left(% \cos\theta\right)+\mathsf{Q}^{\mu}_{\nu}\left(\cos\theta\right)=\pi^{\frac{1}{% 2}}\Gamma\left(\nu+\mu+1\right)(2\sin\theta)^{\mu}e^{\pm(\nu+\mu+1)i\theta}\*% \mathbf{F}\left(\nu+\mu+1,\mu+\frac{1}{2};\nu+\frac{3}{2};e^{\pm 2i\theta}% \right).}}
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    DLMF:14.13#Ex1
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